Construction of Graphs and Zero-Divisor Graphs by Degree-Joint Operations of Graphs

The new construction of graphs obtained by degree-joint operations between maximum-degree and minimum-degree of some types of graphs with different vertex and edge sets denoted by Gδ,GΔ respectively, joins vertices of maximum (minimum) degree in one graph with those of the same type in the other graph of the second graph, where the degree-joint operation is defined as minimum–maximum or maximum–minimum degree, denoted by G∇. In the current paper, we have examined the important properties of these graphs obtained by a new process and concluded that if one of the graphs G1 or G2 is regular, then GΔ, Gδ, and G∇ are n+mG-regular. Obviously, the new graphs are connected if one of the components is connected, with a diameter less than the sum of the diameters of the component graphs G1 and G2. Finally, we find some important properties of the maximum-degree join of the zero-divisor graph of local rings and the direct product of two finite fields.